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Compositional Logical Semantics Torbjörn Lager Department of Linguistics Stockholm University
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NLP1 - Torbjörn Lager 2 Logical Semantics Example zJohn laughed zlaughed'(j) zNobody laughed x[laughed'(x)] zBut this is just translation! What's semantic about that?
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NLP1 - Torbjörn Lager 3 What Is the Name of This Game? zTruth conditional semantics zModel theoretic semantics zLogical semantics zFormal semantics zCompositional semantics zSyntax-driven semantic analysis zCompositional, logical, truth conditional, model theoretic semantics....
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NLP1 - Torbjörn Lager 4 Truth Conditional Semantics zWe use language to talk about the world zMeaning = Truth conditions zExamples: y"John whistles" is true iff John whistles y"John visslar" is true iff John whistles y"Ogul fautu seq" is true iff... Natural language The outside world
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NLP1 - Torbjörn Lager 5 Model Theoretic Semantics zWe don't know what the world is really like, so let's talk about a model of the world instead zSuch a model does (usually) consists of individuals, sets of individuals, functions and relations. i.e the sort of things set theory talks about zTruth becomes truth relative to a model Natural language Model The outside world
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NLP1 - Torbjörn Lager 6 Compositional Semantics zThe Compositionality Principle: yThe meaning of the whole is a function of the meaning of the parts and the mode of combining them. yThe meaning of a complex expression is uniquely determined by the meaning of its constituents and the syntactic construction used to combine them. Natural language Model The World
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NLP1 - Torbjörn Lager 7 Truth Conditional, Model Theoretic and Compositional Semantics Combined zA simple model M: yDomain: x{John, Mary, Paul} yInterpretation: xNames: "John" refers to John, "Mary" refers to Mary, etc. xVerbs: "whistles" refers to {John, Paul} zExample y"John whistles" is true in M iff the individual in M referred to as "John" is an element in the set of individuals that "whistles" refer to. Richard Montague (1970): "I reject the contention that an important theoretical difference exists between formal and natural languages"
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NLP1 - Torbjörn Lager 8 Translational Semantics zAccount for the meanings of natural language utterances by translating them into another language. zIt could be any language, but only if this language has a formal semantics are we done. Natural language Logical Form Language Model The World
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NLP1 - Torbjörn Lager 9 Grammar and Logical Form zS -> NP VP [S] = [VP]([NP]) zNP -> john [NP] = j zVP -> whistles [VP] = x[whistles'(x)] z[john whistles] = whistles'(j) cf. The principle of compositionality Have same truth conditions
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NLP1 - Torbjörn Lager 10 Beta Reduction (Lambda Conversion) z[S] = [VP]([NP]) z[NP] = j z[VP] = x[whistles'(x)] Beta reduction rule : u ( ) where every occurrence of u in is replaced by z x[whistles'(x)](j) application zwhistles'(j) reduction
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NLP1 - Torbjörn Lager 11 Grammar and Logical Form zS -> NP VP [S] = [NP]([VP]) zNP -> john [NP] = P[P(j)] zVP -> whistles [VP] = x[whistles'(x)] z[john whistles] = whistles'(j)
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NLP1 - Torbjörn Lager 12 From Logical Form to Truth Conditions whistles'(j) is true iff the individual (in the model) denoted by 'j' has the property denoted by 'whistles' cf. "John whistles" is true iff John whistles
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NLP1 - Torbjörn Lager 13 Beta Reduction (Lambda Conversion) z[S] = [NP]([VP]) z[NP] = P[P(j)] z[VP] = x[whistles'(x)] Beta reduction rule : u ( ) where every occurrence of u in is replaced by z P[P(j)]( x[whistles'(x)]) application z x[whistles'(x)](j) reduction zwhistles'(j) reduction
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NLP1 - Torbjörn Lager 14 A Larger Example zS -> NP VP [S] = [NP]([VP]) zNP -> DET N [NP] = [DET]([N]) zDET -> every [DET] = Q[ P[ z[Q(z) P(z)]]] zN -> man [N] = x[man'(x)] zVP -> whistles [VP] = x[whistles'(x)] z[every man whistles} = z[man'(z) whistles'(z)]
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NLP1 - Torbjörn Lager 15 A Larger Example (cont'd) z[S] = [NP]([VP]) z[NP] = [DET]([N]) z[DET] = Q[ P[ z[Q(z) P(z)]]] z[N] = x[man'(x)] z[VP] = x[whistles'(x)] z Q[ P[ z[Q(z) P(z)]]]( x[man'(x)]) application z P[ z[ x[man'(x)](z) P(z)]] reduction z P[ z[man'(z) P(z)]] reduction z P[ z[man'(z) P(z)]]( x[whistles'(x)]) application z z[man'(z) x[whistles'(x)](z)] reduction z z[man'(z) whistles'(z)] reduction
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NLP1 - Torbjörn Lager 16 Examples in Oz zX = {fun {$ P} {P j} end fun {$ X} whistles(X) end} z% X gets bound to the tuple 'whistles(j)' zDetSem = fun {$ Q} fun {$ P} all(Z impl({Q Z} {P Z})) end end zNSem = fun {$ X} man(X) end zVPSem = fun {$ X} whistles(X) end zNPSem = {DetSem NSem} zSSem = {NPSem VPSem} z% X gets bound to the tuple 'all(Z impl(man(Z) whistles(Z)))'
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